Abstract

We introduce the notion of h-stability for fractional differential systems. Then we investigate the boundedness and h-stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give examples to illustrate our results.

1. Introductions and Preliminaries

Lakshmikantham et al. [1–5] investigated the basic theory of initial value problems for fractional differential equations involving Riemann-Liouville differential operators of order . They followed the classical approach of the theory of differential equations of integer order in order to compare and contrast the differences as well as the intricacies that might result in development [6, Vol. I]. Li et al. [7] obtained some results about stability of solutions for fractional-order dynamic systems using fractional Lyapunov direct method and fractional comparison principle. Choi and Koo [8] improved on the monotone property of Lemma in [5] for the case with a nonnegative real number . Choi et al. [9] also investigated Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.

In this paper we introduce the notion of -stability for fractional differential equations. Then, we investigate the boundedness and -stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give some examples to illustrate our results.

For the basic notions and theorems about fractional calculus, we mainly refer to some books [5, 10, 11].

We recall the notions of Mittag-Leffler functions which were originally introduced by Mittag-Leffler in 1903 [12]. Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional order systems is the Mittag-Leffler function, defined as
where and is the Gamma function [11]. The Mittag-Leffler function with two parameters has the following form:
where and . For , we have . Also, .

Note that the exponential function possesses the semigroup property (i.e., for all ), but the Mittag-Leffler function does not satisfy the semigroup property unless or [13].

Definition 1 (see [5]). The Riemann-Liouville fractional integral of order of a function is defined as
where (provided that the integral exists in the Lebesgue sense).

Definition 2 (see [5]). The Riemann-Liouville fractional derivative of order of a continuous function is given by
provided that the right side is pointwise defined on .

If , then the Riemann-Liouville fractional derivative of order of a function reduces to

Note that the Riemann-Liouville fractional derivatives have singularity at and the fractional equations in the Riemann-Liouville sense require initial conditions at some point different from . To overcome this issue, Caputo [14] defined the fractional derivative in the following way.

Definition 3 (see [10]). Let be a positive real number such that for . The Caputo fractional derivative of order of a function is defined by
where .

When , then the Caputo fractional derivative of order of reduces to

When , we have
In particular, if , then we have

Hence, we can see that the Caputo derivative is defined for functions for which the Riemann-Liouville derivative exists. Also, we note that the Mittag-Leffler functions and satisfy the more general differential relations
respectively, for .

We can obtain the following asymptotic property for and from the result [10, page 51].

Lemma 4 (see [10]). When , then has different asymptotic behavior at infinity for and . (1)If and is a real number such that
then, for , the following asymptotic expansions are valid:
with ; and
with .(2)When , then, for , the following asymptotic estimate holds:
with , and where the first sum is taken over all integer such that

Lemma 5. Let and . Then, and tend monotonically to zero as .

Proof. If we set and in Lemma 4, then it follows from Lemma 4 that for we have
Thus, we have
For , we also have
by the above similar argument. This completes the proof.

Corollary 6. Let and . Then, one has

2. Main Results

Let and . Denote by the function space

Let be a domain and . We consider the Caputo fractional differential system with the initial value
where . If satisfies (21), it also satisfies the Volterra fractional integral equation
and vice versa.

In the sequential we assume that the solution of (21) exists globally on . See [5, Theorem ] for the existence and uniqueness result.

Next, we consider the nonhomogeneous linear fractional differential equation with Caputo fractional derivative
where is Hölder continuous with exponent . Then, we get the unique solution of (23) as
for each .

Lemma 7 (see [9, Lemma 3.2]). If one sets in (23) with a constant , then the solution of (24) reduces to

Remark 8. If , then it follows from Lemma 7 that
We can obtain the following Caputo fractional differential inequality of Gronwall type by Lemma 7.

Lemma 9. Suppose that satisfies
where . Then one has

Proof. There exists a nonnegative function satisfying
It follows from Lemma 7 that
where denotes the convolution operator of nonnegative functions and . Since is nonnegative for each , then we have
This completes the proof.

Remark 10. If we set and in Lemma 9, then we have
We can obtain the following result about fractional integral inequality. It is adapted from the comparison principle regarding nonstrict inequalities in [2, 5].

Lemma 11 (see [8, Lemma 2.11]). Let and . Suppose that satisfy the fractional integral inequality:
where and is monotonic nondecreasing in for each . If , then one has on .

Pinto [15] introduced -stability which is an important extension of the notions of exponential stability and uniform Lipschitz stability for differential equations.

We will give the notion of -stability for Caputo fractional differential systems.

Definition 12. The zero solution of (21) is said to be (1)an -system if there exist a constant and a positive continuous function such that
for . Here .(2)-stable if is bounded.

Definition 13. The zero solution of (21) is said to be a Mittag-Leffler system if
where , and are locally Lipschitz on with Lipschitz constant .The zero solution of (21) is called Mittag-Leffler stable if the constant in (35) is nonpositive.

Note that the Mittag-Leffler stability implies -stability, but the converse does not hold in general. See Remark 19 for the example.

Theorem 14. Suppose that the function of (21) satisfies
where is monotonic increasing in for each with . One considers the Caputo fractional differential equation
If the zero solution of (37) is an -system, then the zero solution of (21) is also an -system whenever .

Proof. The equation (21) is equivalent to the following Volterra fractional integral equation:
Then, we obtain
Thus we have
where . By Lemma 11, we have for all . Since of (37) is an -system, there exist a constant and a positive continuous function such that
for . Thus, we see that
where with and . This completes the proof.

Corollary 15. Suppose that all conditions of Theorem 14 hold. The asymptotic stability of (37) implies the corresponding asymptotic stability of (21).

We can obtain an upper bound of solutions for Caputo fractional differential equations via fractional Gronwall’s inequality. The following result is adapted from Theorem 5.1 in [7] and Theorem 3.15 in [9].

Lemma 16. Suppose that is a domain containing the origin and . Let be a continuously differentiable function and locally Lipschitz with respect to satisfying
where and are positive constants. Then one has
where is any solution of (21).

Proof. It follows from (43) and (44) that
It follows from Lemma 9 that
Substituting (47) into (43) yields
This complete the proof.

We can obtain the boundedness of solutions for Caputo fractional differential equations via the fractional Lyapunov direct method.

Theorem 17. Under the same assumptions of Lemma 16, all solutions of (21) are eventually bounded on .

Proof. Let be any solution of (21). Then it follows from Lemma 16 that
for each . In view of Lemma 5 and Corollary 6, we note that tends monotonically zero as and is eventually bounded on . Hence, there exist a positive constant and such that
This completes the proof.

We can obtain the following result [7, Theorem 5.1] about Mittag-Leffler stability of (21) as a corollary of Lemma 16.

Corollary 18. If one sets in the assumption of Lemma 16, then the zero solution of (21) is Mittag-Leffler stable.

3. Examples

In this section we give tow examples which illustrate some results in the previous section.

Example 1 (see [8]). To illustrate Theorem 14, we consider the Caputo fractional differential equation
where . Then the zero solution of (51) is -stable.

Proof. The function satisfies
and the solution of the Caputo fractional differential equation
is given by . We have
where . Thus, the zero solution of (53) is -stable. Hence, the zero solution of (51) is -stable by Theorem 14.

Remark 19. We note that the fractional differential equation (53) given in the proof of Example 1 is -stable but not Mittag-Leffler stable.

Proof. Let be any solution of (53). Then, it follows from [8, Example 2.2] that is neither monotonic nondecreasing in nor monotonic nonincreasing in . Furthermore, we easily see that
Suppose that (53) is Mittag-Leffler stable; that is, there exist positive constants and satisfying
where , and is locally Lipschitz in . Since is monotonic nonincreasing in [17], we see that the right-hand function of (56) also is monotonic decreasing in . This contradicts the fact that has neither monotonic nondecreasing property nor monotonic nonincreasing property.

Example 2. Let . We consider the Caputo fractional differential equation
where is bounded by a constant . Let . Then it follows that
Let . Applying them in Lemma 16 gives
Hence, all solutions of (57) are eventually bounded by Theorem 17.

Conflict of Interests

The authors declare no conflict of interests.

Acknowledgment

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2007585).